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Non-Technical SummaryWhat is the best approach to modeling money? This question is one of those that economists have struggled with for a while, and is not yet settled.In this paper, we juxtapose two microfounded monetary models' equilibrium predictions and their quantitative implications for the welfare cost of inflation: the model in Lagos and Wright (2005) (hereafter LW) and the cash-in-advance model in tradition of Lucas (1980Lucas ( , 1984. Proponents of the LW model view it as being theoretically more appealing than reduced-form models such as cashin-advance, and have argued that the model can generate significantly different quantitative results; e.g., the welfare costs of inflation are higher than in reduced-form models.The two models exhibit key similarities. In both models agents synchronously alternate between a centralized market and a decentralized market; consumption utility depends on the market in which the purchase is settled; adjustments of money balances are made before a random shock is observed. A key difference is the use of Nash bargaining in one model, but not the other: in the LW model, bargaining determines prices of cash trades, which induces a price distortion depending on the seller's bargaining power. Do these theoretical platforms predict different equilibrium allocations? Are these models generally incapable of producing similar quantitative results? What model features are responsible for possible disparities?Once the two frameworks are placed on equal footing, in terms of preferences, technologies, and shocks, we find that the equations characterizing stationary equilibrium in the LW model when sellers have no bargaining power coincide with the equations that characterize stationary competitive equilibrium in the cash-in-advance model. This also holds if sellers do have some bargaining power, when the price distortion from Nash bargaining is replicated in the other model via a tax on cash revenues. Such correspondence between equations immediately extends outside of steady-state, if sellers have no bargaining power and workers have isoelastic preferences; otherwise, the equations do not generally correspond. A quantitative exercise demonstrates that the welfare costs of inflation in the cash-in-advance model match those in the LW model.The message is that differences in the models' main equations reduce to differences in the pricing mechanism assumed to govern those transactions that must be se...